I How to recalculate Berry phase dipole moments (jumps over time)?

[bumblebee77]

TL;DR Summary

I have Berry phase dipole moments over time from a simulation but don't understand how to correct the jumps in the data.

Is there anyone out there who can help a newbie understand how to deal with Berry phase dipole moment data?

I ran a simulation to calculate dipole moments over time using the Berry phase method. The problem is that there are jumps in my data. There is an example at the end of this post (part of the z component data), where one row (time step) is +23.x and the next row is -23.x.

For each dipole moment (row), I have this:
Dipole vectors are modulo integer multiples of the cell matrix.
[X] [ 45.8125 0.0000 0.0000 ] [i ]
[Y]=[ 0.0000 45.8125 0.0000 ]*[j]
[Z] [ 0.0000 0.0000 45.8125 ] [k]

I think what is going on is that in the Berry Phase, the dipole moment is not an absolute value. It is tied to the cell matrix. So when a dipole moment component exceeds the value in the cell matrix (45.8125 in my example), then it shows up in my output as some number that represents 45.8125 --> reset to 0--> add whatever is left.

Could someone please let me know if I am on the right track? I am trying to figure out how to correct my output dipole moment data to get a time series that I can use in a calculation. If anyone knows how to do that and could explain or point me to an online resource, I would be very appreciative.

22.11478446
22.33207588
23.86209392
-23.4624732
-23.04423182
-22.6735481
-22.36681344
-22.13680091
-21.99437418
-21.94794955
-22.00093276
-22.14894473
-22.37851464
-22.98852983
-23.30787693
-23.59047008
-23.80075739
23.86218945
-23.88415599
-23.72229234
-23.42677872
-23.02231253
-22.55223629
-22.07417372
-21.65136982
-21.34266766
-21.19465513

 

[CarlB]

Hmmm. Isn't Berry phase a phase and so you can add multiples of 2 pi to it? So figure out what amount a Berry phase of 2 pi is, when converted to a dipole moment, and add or subtract that amount. This is the "polarization quantum" and I suppose it is related to the unit cell volume.

 

[bumblebee77]

Thank you, CarlB. Yes, I read that the Berry phase is about periodic systems and that the 2*pi is related to that. So I expect that if I want to use my simulation output, I will have to correct the data.

I don't know whether the 2*pi is relevant here though since we have the cell size at every time step.

My issue is that I have no idea how to know when to add/subtract anything to my output.

 

[CarlB]

Any time you are computing a Berry phase you always have an unknown 2 pi just as when you measure an angle in the real world you cannot distinguish between, say -pi and +pi, or between -pi/2 and 3 pi/2.

Let me put it this way. So if you had a computer computing angles in the real world you would also end up with sudden jumps of 2 pi for the same reason. You need to adjust them to stay on the same branch. Let me look for a reference...

Try section (3.3) of this fairly easy to read reference:
"It is clear that (22), being a phase, is only well-defined modulo 2π. We can
see this more explicitly in (21); let ..."
http://www.physics.rutgers.edu/~dhv/pubs/local_preprint/dv_fchap.pdf

 

[CarlB]

By the way, where I end up using Berry phase the most is when a spin-1/2 ket is sent around a closed path on the Bloch sphere. For example, suppose it starts at +z, then +x, then to +y then back to +z. What is the Berry phase for this path?

Well it is half the surface area of the sphere cut out by the path. In this case the path takes 1/8 (an octant) of the sphere so the Berry phase is 4 pi x 1/8 x 1/2 = pi/4. The 4 pi is the surface area of a unit sphere. 1/8 is the amount the path encloses and the 1/2 factor is an adjustment that I think of as because the surface area corresponds to 2 pi Berry phase but has 4 pi area. Using the kets
|+z> = (1,0),
|+x> = (1,1),
|+y> = (1,i),
|+z> = (1,0)
where I've left off some sqrt(1/2) because we're only interested in phases and not magnitudes here. Then we compute the phase using a product of density matrices:
|+z><+z|+y><+y|+x><+x|+z><+z| = |+z> 1 (1-i) 1<+z| = (1-i)|+z><+z|
and the density matrix has been multiplied by (1-i). Putting the sqrt(1/2)s back in the actual multiplication is (1-i)/2 = exp(-i pi/4) and so the Berry phase = -pi/4 as expected, but the path is apparently in the negative direction so we had an overall minus sign.

Now in the above calculation I didn't have any 2 pi problems but I suppose that the reason is that I did it with density matrices which do not have arbitrary complex phases. And if I continuously modify the path I'll stay on the same branch so no 2 pi jumps will occur. So I'm wondering if there is a density matrix method of doing the calculation you're doing that avoids the 2 pi problem. I'm not going to guess about that but I'd love to hear what you think about it.

 

[bumblebee77]

Thank you so much, Carl. Very kind of you to explain. It's going to take me a while to digest this! Maybe a day or two. I will be back!

Regarding the density matrix, I do suspect that's what this from my original question is about because (in my very basic thinking) there is no mention of pi anywhere.

"For each dipole moment (row), I have this:
Dipole vectors are modulo integer multiples of the cell matrix.
[X] [ 45.8125 0.0000 0.0000 ] [i ]
[Y]=[ 0.0000 45.8125 0.0000 ]*[j]
[Z] [ 0.0000 0.0000 45.8125 ] [k]"